CDF of Berk-Jones statitic under the null hypothesis.

Description

CDF of Berk-Jones statitic under the null hypothesis.

Usage

pbj(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

The left-tail probability of the null distribution of B-J statistic at the given quantile.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

See Also

stat.bj for the definition of the statistic.

Examples

pval <- runif(10)
bjstat <- stat.phi(pval, s=1, k0=1, k1=10)$value
pbj(q=bjstat, M=diag(10), k0=1, k1=10)

CDF of Higher Criticism statistic under the null hypothesis.

Description

CDF of Higher Criticism statistic under the null hypothesis.

Usage

phc(q, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

The left-tail probability of the null distribution of HC statistic at the given quantile.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

See Also

stat.hc for the definition of the statistic.

Examples

pval <- runif(10)
hcstat <- stat.phi(pval, s=2, k0=1, k1=5)$value
phc(q=hcstat, M=diag(10), k0=1, k1=10)

Statistical power of Berk and Jones test.

Description

Statistical power of Berk and Jones test.

Usage

power.bj(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

Details

We consider the following hypothesis test,

H_0: X_i\sim F, H_a: X_i\sim G

Specifically, F = F_0 and G = (1-\epsilon)F_0+\epsilon F_1, where \epsilon is the mixing parameter, F_0 and F_1 is speified by the "method" argument:

"gaussian-gaussian": F_0 is the standard normal CDF and F = F_1 is the CDF of normal distribution with \mu defined by mu and \sigma = 1.

"gaussian-t": F_0 is the standard normal CDF and F = F_1 is the CDF of t distribution with degree of freedom defined by df.

"t-t": F_0 is the CDF of t distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central t distribution with degree of freedom defined by df and non-centrality defined by delta.

"chisq-chisq": F_0 is the CDF of Chisquare distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

"exp-chisq": F_0 is the CDF of exponential distribution with parameter defined by df and F = F_1 is the CDF of non-central Chisqaure distribution with degree of freedom defined by df and non-centrality defined by delta.

Value

Power of BJ test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

3. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

4. Berk, R.H. & Jones, D.H. Z. "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Wahrscheinlichkeitstheorie verw Gebiete (1979) 47: 47.

See Also

stat.bj for the definition of the statistic.

Examples

power.bj(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)

Statistical power of Higher Criticism test.

Description

Statistical power of Higher Criticism test.

Usage

power.hc(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

Details

We consider the following hypothesis test,

H_0: X_i\sim F, H_a: X_i\sim G

Specifically, F = F_0 and G = (1-\epsilon)F_0+\epsilon F_1, where \epsilon is the mixing parameter, F_0 and F_1 is speified by the "method" argument:

"gaussian-gaussian": F_0 is the standard normal CDF and F = F_1 is the CDF of normal distribution with \mu defined by mu and \sigma = 1.

"gaussian-t": F_0 is the standard normal CDF and F = F_1 is the CDF of t distribution with degree of freedom defined by df.

"t-t": F_0 is the CDF of t distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central t distribution with degree of freedom defined by df and non-centrality defined by delta. "chisq-chisq": F_0 is the CDF of Chisquare distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

"exp-chisq": F_0 is the CDF of exponential distribution with parameter defined by df and F = F_1 is the CDF of non-central Chisqaure distribution with degree of freedom defined by df and non-centrality defined by delta.

Value

Power of HC test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

See Also

stat.hc for the definition of the statistic.

Examples

power.hc(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)

Statistical power of phi-divergence test.

Description

Statistical power of phi-divergence test.

Usage

power.phi(
  alpha,
  n,
  s,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Arguments

Details

We consider the following hypothesis test,

H_0: X_i\sim F, H_a: X_i\sim G

Specifically, F = F_0 and G = (1-\epsilon)F_0+\epsilon F_1, where \epsilon is the mixing parameter, F_0 and F_1 is speified by the "method" argument:

"gaussian-gaussian": F_0 is the standard normal CDF and F = F_1 is the CDF of normal distribution with \mu defined by mu and \sigma = 1.

"gaussian-t": F_0 is the standard normal CDF and F = F_1 is the CDF of t distribution with degree of freedom defined by df.

"t-t": F_0 is the CDF of t distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central t distribution with degree of freedom defined by df and non-centrality defined by delta.

"chisq-chisq": F_0 is the CDF of Chisquare distribution with degree of freedom defined by df and F = F_1 is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

"exp-chisq": F_0 is the CDF of exponential distribution with parameter defined by df and F = F_1 is the CDF of non-central Chisqaure distribution with degree of freedom defined by df and non-centrality defined by delta.

Value

Power of phi-divergence test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

See Also

stat.phi for the definition of the statistic.

Examples

#If the alternative hypothesis Gaussian mixture with eps = 0.1 and mu = 1.2:#
power.phi(0.05, n=10, s=2, beta=0.5, eps = 0.1, mu = 1.2)

calculate the left-tail probability of phi-divergence under general correlation matrix.

Description

calculate the left-tail probability of phi-divergence under general correlation matrix.

Usage

pphi(q, M, k0, k1, s = 2, t = 30, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

Left-tail probability of the phi-divergence statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

Examples

M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
pphi(q=2, M=M, k0=1, k1=5, s=2)
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ecc")
pphi(q=2, M=M, k0=1, k1=5, s=2, method = "ave")
pphi(q=2, M=diag(10), k0=1, k1=5, s=2)

calculate the left-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the left-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Usage

pphi.omni(q, M, K0, K1, S, t = 30, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

Left-tail probability of omnibus phi-divergence statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

Examples

M = matrix(0.3,10,10) + diag(1-0.3, 10)
pphi.omni(0.05, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))

Quantile of Berk-Jones statistic under the null hypothesis.

Description

Quantile of Berk-Jones statistic under the null hypothesis.

Usage

qbj(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)

Arguments

Value

Quantile of BJ statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

See Also

stat.bj for the definition of the statistic.

Examples

## The 0.05 critical value of BJ statistic when n = 10:
qbj(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qbj(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)

Quantile of Higher Criticism statistics under the null hypothesis.

Description

Quantile of Higher Criticism statistics under the null hypothesis.

Usage

qhc(p, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL, err_thr = 1e-04)

Arguments

Value

Quantile of HC statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

See Also

stat.hc for the definition of the statistic.

Examples

## The 0.05 critical value of HC statistic when n = 10:
qhc(p=.95, M=diag(10), k0=1, k1=5, onesided=FALSE)
qhc(p=1-1e-5, M=diag(10), k0=1, k1=5, onesided=FALSE, err_thr=1e-8)

Quantile of phi-divergence statistic under the null hypothesis.

Description

Quantile of phi-divergence statistic under the null hypothesis.

Usage

qphi(
  p,
  M,
  k0,
  k1,
  s = 2,
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL,
  err_thr = 1e-04
)

Arguments

Value

Quantile of the phi-divergence statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

See Also

stat.phi for the definition of the statistic.

Examples

qphi(p=.95, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-3, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-6)
qphi(p=1-1e-5, M=diag(10), k0=1, k1=5, s=2, onesided=FALSE, err_thr = 1e-8)

Construct Berk and Jones (BJ) statistics.

Description

Construct Berk and Jones (BJ) statistics.

Usage

stat.bj(p, k0 = 1, k1 = NA)

Arguments

Details

Let p_{(i)}, i = 1,...,n be a sequence of ordered p-values, the Berk and Jones statistic

BJ = \sqrt{2n} \max_{1 \leq i\leq \lfloor \beta n \rfloor} (-1)^j \sqrt{i/n * \log(i/n/p_{(i)}) + (1-i/n) * \log((1-i/n)/(1-p_{(i)}))}

and when p_{(i)} > i/n, j = 1, otherwise j = 0.

Value

value - BJ statistic constructed from a vector of p-values.

location - the order of the p-values to obtain BJ statistic.

stat - vector of marginal BJ statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

3. Berk, R.H. & Jones, D.H. Z. "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Wahrscheinlichkeitstheorie verw Gebiete (1979) 47: 47.

Examples

stat.bj(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.bj(p.test, k0 = 2, k1 = 20)

Construct Higher Criticism (HC) statistics.

Description

Construct Higher Criticism (HC) statistics.

Usage

stat.hc(p, k0 = 1, k1 = NA)

Arguments

Details

Let p_{(i)}, i = 1,...,n be a sequence of ordered p-values, the higher criticism statistic

HC = \sqrt{n} \max_{1 \leq i\leq \lfloor \beta n \rfloor} [i/n - p_{(i)}] /\sqrt{p_{(i)}(1 - p_{(i)})}

Value

value - HC statistic constructed from a vector of p-values.

location - the order of the p-values to obtain HC statistic.

stat - vector of marginal HC statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

Examples

stat.hc(runif(10))
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.hc(p.test, k0 = 1, k1 = 10)

Construct phi-divergence statistics.

Description

Construct phi-divergence statistics.

Usage

stat.phi(p, s, k0 = 1, k1 = NA)

Arguments

Details

Let p_{(i)}, i = 1,...,n be a sequence of ordered p-values, the phi-divergence statistic

PHI = \sqrt{2n}/(s - s^2) \max_{1 \leq i\leq \lfloor \beta n \rfloor} (-1)^j \sqrt{1 - (i/n)^s (p_{(i)})^s - (1-i/n)^{(1-s)} * (1-p_{(i)})^{(1-s)}}

and when p_{(i)} > i/n, j = 1, otherwise j = 0.

Value

value - phi-divergence statistic constructed from a vector of p-values.

location - the order of the p-values to obtain phi-divergence statistic.

stat - vector of marginal phi-divergence statistics.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

Examples

stat.phi(runif(10), s = 2)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 1 - pnorm(stat.test)
stat.phi(p.test, s = 0.5, k0 = 2, k1 = 5)

calculate the omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the omnibus phi-divergence statistics under general correlation matrix.

Usage

stat.phi.omni(
  p,
  M,
  K0 = rep(1, 2),
  K1 = rep(length(M[1, ]), 2),
  S = c(1, 2),
  t = 30,
  onesided = FALSE,
  method = "ecc",
  ei = NULL
)

Arguments

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

Examples

p.test = runif(10)
M = toeplitz(1/(1:10)*(-1)^(0:9)) #alternating polynomial decaying correlation matrix
stat.phi.omni(p.test, M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))

Multiple comparison test using Berk and Jones (BJ) statitics.

Description

Multiple comparison test using Berk and Jones (BJ) statitics.

Usage

test.bj(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

pvalue - the p-value of the Berk-Jones test.

bjstat - the Berk-Jones statistic.

location - the order of the input p-values to obtain BJ statistic.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668. 4. Leah Jager and Jon Wellner. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

See Also

stat.bj for the definition of the statistic.

Examples

test.bj(runif(10), M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.bj(p.test, M=diag(20), k0=1, k1=10)

Multiple comparison test using Higher Criticism (HC) statitics.

Description

Multiple comparison test using Higher Criticism (HC) statitics.

Usage

test.hc(prob, M, k0, k1, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

pvalue - The p-value of the HC test.

hcstat - HC statistic.

location - the order of the input p-values to obtain HC statistic.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668. 4. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

See Also

stat.hc for the definition of the statistic.

Examples

pval.test = runif(10)
test.hc(pval.test, M=diag(10), k0=1, k1=10)
#When the input are statistics#
stat.test = rnorm(20)
p.test = 2*(1 - pnorm(abs(stat.test)))
test.hc(p.test, M=diag(20), k0=1, k1=10)

Multiple comparison test using phi-divergence statistics.

Description

Multiple comparison test using phi-divergence statistics.

Usage

test.phi(prob, M, k0, k1, s = 2, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

pvalue - The p-value of the phi-divergence test.

phistat - phi-diergence statistic.

location - the order of the input p-values to obtain phi-divergence statistic.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668. 4. Leah Jager and Jon Wellner. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

See Also

stat.phi for the definition of the statistic.v

Examples

stat.test = rnorm(20) # Z-scores
p.test = 2*(1 - pnorm(abs(stat.test)))
test.phi(p.test, M=diag(20), s = 0.5, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 1, k0=1, k1=10)
test.phi(p.test, M=diag(20), s = 2, k0=1, k1=10)

calculate the right-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Description

calculate the right-tail probability of omnibus phi-divergence statistics under general correlation matrix.

Usage

test.phi.omni(prob, M, K0, K1, S, onesided = FALSE, method = "ecc", ei = NULL)

Arguments

Value

p-value of the omnibus test.

p-values of the individual phi-divergence test.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and power of optimal signal-detection statistics in finite case", IEEE Transactions on Signal Processing (2020) 68, 1021-1033 2. Hong Zhang and Zheyang Wu. "The general goodness-of-fit tests for correlated data", Computational Statistics & Data Analysis (2022) 167, 107379 3. Hong Zhang and Zheyang Wu. "Generalized Goodness-Of-Fit Tests for Correlated Data", arXiv:1806.03668.

Examples

M = matrix(0.3,10,10) + diag(1-0.3, 10)
test.phi.omni(runif(10), M=M, K0=rep(1,2), K1=rep(5,2), S=c(1,2))